1. Technical Field
The present invention relates to a method and a computer readable media for determining orientation of fibers in a fluid.
2. Background
Man-made fiber-reinforced thermoplastic (FRT) composites are widely-known materials in polymer engineering technology, with various techniques of injection molding and extrusion being commonly used. The advantages of light weight, high performance, low production cost and good recyclability make FRT composites preferred options to replace metals and thermoset composites. Such FRT materials have improved mechanical properties, including tensile property, thermal expansion, electrical conductivity, and water penetrability.
FRT composites are grouped into two categories based on fiber length: short fiber-reinforced thermoplastics or SFRTs, with fiber length of about 0.2 to 0.4 mm, and long fiber-reinforced thermoplastics or LFRTs, having fiber length of about 10 to 13 mm. Unlike SFRTs, LFRTs can yield continuous-fiber reinforcement. LFRT pellets are more extensively employed in automotive industrial fabrication than SFRT pellets.
FIG. 1 illustrates an injection molding apparatus 10 and a mold unit 20 according to the prior art. The injection molding apparatus 10 includes a screw chamber 11, a heating element 13 configured to heat the screw chamber 11, and a screw 15 positioned in the screw chamber 11. The mold unit 20 includes a sprue 21, a runner 23 and cavities 25. The injection molding apparatus 10 is configured to inject molding material into the cavity 25 of the mold unit 20. The injection molding technique uses conventional rapid automated molding equipment. SFRT/LFRT production has been applied using the injection process. In the injection process, the additional fiber composites filled in polymer/resin melts are transported as a suspension into the mold cavity 25. To design products effectively, the influence of flow-induced fiber orientation distribution on the properties of the finished part must be considered. The effect of divergent/convergent channels of the mold cavity 25 on the alignment of the fibers 30 is clearly illustrated in FIG. 2A and FIG. 2B.
FIG. 3 illustrates the orientation of fibers in the mold cavity 25. The most noticeable feature of the filling is the existence of a shell region 31 and a core region 33 across the thickness of the molded cavity 25. The fibers found in the shell region 31 (near the cavity wall) are strongly aligned in the flow direction, but the fibers in the core region 33 (near the cavity center) are transverse to the flow. Hence, it is necessary to understand how the fiber orientation varies during the mold filling.
Fiber suspension rheology is a highly developed and crucial component of processing technology. Fiber orientation technology is based on Jeffery's time evolution equation for the motion of an isolated rigid spheroid immersed in a Newtonian fluid (see, G. B. Jeffery, “The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid,” Proc. R. Soc. A 102, 161 (1922)). The significant orientation of the Jeffery theory is due to the flow-induced hydrodynamics in dilute suspension.
For a concentrated fiber suspension, Folgar and Tucker (see, F. Folgar and C. L. Tucker III, Orientation Behavior of Fibers in Concentrated Suspensions, J. Reinf. Plast. Compos. 3, 98 (1984)) construed an Isotropic Rotary Diffusion Model that was essentially added to Jeffery's equation of the fiber hydrodynamic term. Subsequently, Advani and Tucker (see, S. G. Advani and C. L. Tucker III, “The use of tensors to describe and predict fiber orientation in short fiber composites,” J. Rheol. 31, 751 (1987)) suggested a second moment tensor, averaged over probability space, to describe the orientation state in surface spherical coordinates. Thus, the well-known Folgar-Tucker time-evolution equation of the orientation tensor, which includes both the Jeffery hydrodynamics and the isotropic rotary diffusion terms, is obtained. The Folgar-Tucker orientation equation family has been implemented in almost all commercial software programs for injection molding.
Under recent rheological experiments of transient shear flow for short fiber suspension, FIG. 4 shows the flow directional orientation against shear strain for the measured orientation data (see, A. P. R. Eberle, D. G. Baird, P. Wapperom, and G. M. Vélez-García, “Using transient shear rheology to determine material parameters in fiber suspension theory”, J. Rheol. 53, 685 (2009)) and the orientation prediction by the Folgar-Tucker (FT) Equation. As focusing in a transient region (about strain <50), a critical problem is that the predicted transient rate of orientation is quicker than the experimental measurements. Hence, it is necessary to develop am improvement of the FT Equation for acceptably predicting fiber orientation (see, J. Wang, J. F. O'Gara, and C. L. Tucker III, An Objective Model for Slow Orientation Kinetics in Concentrated Fiber Suspensions: Theory and Rheological Evidence, J. Rheol. 52, 1179 (2008)).
For the prediction of fiber orientation in injection molding, the fiber-reinforced parts have become standard components of the prediction calculation. Previously, Tucker and Huynh (see, C. L. Tucker III and H. M. Huynh, “Fiber orientation in short flow length parts: limitations of current predictions,” presented at the Proceedings of the 17th Annual Meeting of the Polymer Processing Society, Montreal, Quebec, Canada, 21-24 May, 2001) reported that significant differences between the Folgar-Tucker prediction and experimental observations arise when the flow length of injection molding is short (approximate length/thickness <50). That is, the thickness of the core in practice is found to be greater than the predicted thickness of the core. This is not an acceptable prediction.
The urgent problem above was resolved by Tucker, Wang, and O'Gara, who proposed an innovative model, the Reduced Strain Closure (RSC) Model (see, C. L. Tucker III, J. Wang, and J. F. O'Gara, “Method and article of manufacture for determining a rate of change of orientation of a plurality of fibers disposed in a fluid”, U.S. Pat. No. 7,266,469 B1 (2007)). Significantly, Tucker et al. employed a scalar factor κ to reduce the rate of eigenvalues of orientation tensor while the rate of eigenvectors is unchanged. The RSC Model raises two necessary fourth order tensors, consisting of the eigenvalues and the eigenvectors. The scalar factor κ effectively reduces the rate of change of fiber orientation due to the deformation and the diffusion of the fibers and the fluid, but does not affect the rigid body rotation. Thus, the RSC Model is confirmed to have frame indifference objectivity for a wide class of flows.
The studies mentioned above are suited only to short fibers. In equilibrium states, the short fiber orientation is assumed to be random isotropic, but the long fiber orientation is anisotropic. More Recently, Phelps and Tucker (see, J. H. Phelps and C. L. Tucker III, “An anisotropic rotary diffusion model for fiber orientation in short- and long-fiber thermoplastics,” J. Non-Newtonian Fluid Mech. 156 165 (2009)) proposed a two-dimensional diffusion tensor in surface spherical coordinates to obtain an anisotropic rotary diffusion (ARD) orientation equation for long fibers.
For long fiber composites, therefore, the ARD Model, combined with the RSC Model, is referred to as an ARD-RSC Model with six parameters. The ARD-RSC Model has been implemented in commercial software programs for injection molding. Unfortunately, the ARD-RSC Model is not easy to apply because the parameters are so sensitive and cannot be exactly determined. Inappropriate parameter values may cause poor divergence results. No logistics criterion or experiential rule to adjust the six parameters has been proposed, yet. Nevertheless, Phelps and Tucker further offered a program to fit experimental data and obtained many possible sets of parameters. However, such a program is somewhat complicated and inconvenient to use.
Furthermore, the fact that polymer fluids flow in the injection molding process is considered. After being transported into mold cavity and then cease, the fluids are suddenly solidified via cooling system. According to polymer rheology, the rested fluids' velocity vector and the velocity gradient tensor (L) go to zero, while their vorticity tensor (W) and rate-of-deformation tensor (D) to be zero is due to W=½(L−Lt) and D=½(L+Lt), wherein Lt is a transpose matrix of L.
At this point, such fluids consisting of polymer chains should exist in non-equilibrium thermodynamics states and possess residual stresses for themselves. With respect to time, the stress is spontaneously released, up to the chains approaching a quasi-equilibrium state of random coil. This conforms to a prevalent understanding of polymer viscoelasticity.
Prior to suspension rheological theories, the Jeffery hydrodynamic model {dot over (A)}HDJ and the Folgar-Tucker diffusion model are classic to study regarding changes in orientation states of fibers immersed in fluids. In recent, the RSC Model {dot over (A)}RSC developed by Tucker and coworkers is attended in application of orientation prediction to injection molding. As a premise, these models assume fluids of interest to be the Newtonian fluid, wherein their details are available elsewhere (see, J. Wang, J. F. O'Gara, and C. L. Tucker III, An Objective Model for Slow Orientation Kinetics in Concentrated Fiber Suspensions: Theory and Rheological Evidence, J. Rheol. 52, 1179 (2008)). Regrettably, the significant models have not been attempted to discuss, especially in a rested fluid situation, changes in fiber orientation. Thus, {dot over (A)}HDJ={dot over (A)}DFFT={dot over (A)}RSC=0 go to zero in which the flow ceases and their vorticity tensor, rate-of-deformation tensors, and strain rate all equal to zero. This finding indicates no contribution to the change of fiber orientation. In short, applying these aforementioned models based on the Newtonian fluid assumption, changes in orientation of fibers existing in the rested fluids, particularly when configuration variations of polymer chains is due to relaxation of residual stress, should not have been described.